calculus book - Mathematics and Computer Science

Chapter 2
Numbers
There is a pervasive but incorrect perception that mathematics is the
study of numbers; some mathematicians even joke that the public believes
mathematical research consists of extending multiplication tables
to higher and higher factors. This misapprehension is fostered by school
courses that treat routine calculation as the primary goal of mathematics.
In fact, calculation is a skill, whose relationship to mathematics
analogous to the relationship of spelling to literary composition.
In school, you learned about various kinds of numbers: the counting
(natural) numbers, whole numbers (integers), fractions (rational numbers),
and decimals (real numbers). You may even have been introduced
to complex numbers, or at least to their most famous non-real member,
i, a square root of −1. One goal of this chapter is to (re-)acquaint you
with these sets of numbers, and to present their abstract properties—
the associative, commutative, and distributive laws of arithmetic, properties
of inequalities, and so forth. At the same time, you will see (in
outline) how these sets of numbers are constructed from set theory,
and discover that the way you have learned about numbers so far is
almost purely notational. Nothing about the integers requires base 10
notation, and nothing about the real numbers forces us to use infinite
decimals to represent them. You will learn to view numbers notionally,
in terms of axioms that abstract their properties. The philosophical
question “What is a real number?” will evaporate, leaving behind the
answer “An element of a set that obeys several axioms.”
One misnomer should be dispelled immediately: Though √ 2 is a
“real” number while √ −1 is an “imaginary” number, each of these symbols
represents a mathematical abstraction, and neither has an existence
more or less “real” than the other. No physical quantity can
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